\section{note of 2009.09.19}
\subsection{Renormalization of Gap Equation with 2-Body Quantities}
This can be done formally with a 2 by 2 T matrix, which does not mean much.  In 2-body, the close-channel is treated in 1-dimension Hilbert space. This in my mind is one of the key feature in Feshbach resonance, at least in 2-boy treatment.  It leads to reducing from a formal 2 channel problem to single parameter, s-wave scattering length, $a_s$.  

\subsection{Discussion with Shizhong}
I had a discussion over skype with Shizhong tonight.  He agreed that for a normal Green's function with time-independent Hamiltonian, there is no dependence on the absolute time shift.  And for abnormal one, it should be chemical potential. But in this case, BCS wave function is not exactly eigenstate of the BCS Hamiltonian, and the differentiation of equation \eqref{eq:zhang536} is not exactly chemical potential, as the operators does not connect two eigenstates. 

He suggested me to do the renormalization first with a formal 2 by 2 hamiltonian or T-matrix, and go from there to some simplification to $a_s$ or $\delta_c$.  Maybe my problem is not how to renormalization, but how to reduce the 2 by 2 matrix in 2-body to these observable 2-body quantities.    

He recommend R. Combescot's paper (\cite{rCombescot,rCombescot2}) again. He suggested me to think about the non-zero central-motion close channel molecules.  
